# How to Find the Period of a Function?

A periodic function is a function that repeats itself at regular intervals. In the following step-by-step guide, you will learn how to find the period of a function.

The time interval between two waves is known as a period, while a function that repeats its values at regular intervals or periods is known as a periodic function. In other words, the periodic function is a function that repeats its values after each particular period.

## A step-by-step guide to a periodic function

A function $$y = f (x)$$ is a periodic function in which there exists a positive real number $$P$$ such that $$f (x + P) = f (x)$$, for all $$x$$ belong to real numbers.

The smallest value of a positive real number $$P$$ is called the fundamental period of a function.

This fundamental period of a function is also called the function period in which the function repeats itself.

$$\color{blue}{f(x+P)=f(x)}$$

Note: the sine function is a periodic function with a period of $$2π$$. $$sin(2π + x) = sinx$$.

The periods of some important periodic functions are as follows:

• The period of $$sinx$$ and $$cosx$$ is $$2π$$.
• The period of $$tanx$$ and $$cotx$$ is $$π$$.
• The period of $$secx$$ and $$cosecx$$ is $$2$$.

### Properties of periodic functions

The following features are useful for a deeper understanding of the concepts of periodic function:

• The graph of a periodic function is symmetric and repeats itself along the horizontal axis.
• The domain of the periodic function includes all values of real numbers, and the range of the periodic function is defined for a fixed interval.
• The period of a periodic function against which the period is repeated is equal to the constant over the whole range of the function.
• If $$f (x)$$ is a periodic function with period $$P$$, $$\frac{1}{f(x)}$$ will also be a periodic function with the same fundamental period $$P$$.
• If $$f(x)$$ is a periodic function with a period of $$P$$, then $$f(ax + b)$$ is also a periodic function with a period of $$\frac {P}{|a|}$$.
• If $$f(x)$$ is a periodic function with a period of $$P$$, then $$af(x) + b$$ is also a periodic function with a period of $$P$$.

### Periodic Function – Example 1:

Find the period of the periodic function $$y=sin(4x + 5)$$.

Solution:

The period of $$sinx$$ is $$2π$$, and the period of $$sin(4x + 5)$$ is :

$$\frac{2π}{4}=\frac{π}{2}$$

Therefore, the period of $$sin(4x + 5)$$ is $$\frac{π}{2}$$.

### Periodic Function – Example 2:

Find the period of the periodic function $$y=9 cos(6x + 4)$$.

The period of $$cosx$$ is $$2π$$, and the period of $$9 cos(6x + 4)$$ is:

$$\frac{2π}{6}=\frac{π}{3}$$

Therefore, the period of $$9 cos(6x + 4)$$ is $$\frac{π}{3}$$.

## Exercises for Periodic Function

### Find the period of the function.

1. $$\color{blue}{y= tan3x + sin\frac{5x}{2}}$$
2. $$\color{blue}{y=sec(\pi x-2)}$$
3. $$\color{blue}{y=cot(-(\frac{2\pi}{3})x)}$$
4. $$\color{blue}{\:y=cos\left(-\left(\frac{2}{3}\right)x-\pi \right)}$$
1. $$\color{blue}{4\pi}$$
2. $$\color{blue}{2}$$
3. $$\color{blue}{\frac{3}{2}}$$
4. $$\color{blue}{3\pi}$$

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